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Theorem dchrisum0lem1 27806

Description: Lemma for dchrisum0 27810. (Contributed by Mario Carneiro, 12-May-2016.) (Revised by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)
dchrisum0lem1.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
dchrisum0.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrisum0.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrisum0.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))

**Assertion**
Ref	Expression
dchrisum0lem1	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑚,𝑦, 1 𝑚,𝑑,𝑥,𝑦,𝐶 𝐹,𝑑,𝑥,𝑦 𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝑁,𝑥,𝑦 𝜑,𝑑,𝑚,𝑥 𝑆,𝑑,𝑚,𝑥,𝑦 𝑥,𝑊 𝑚,𝑍,𝑥,𝑦 𝐷,𝑚,𝑥,𝑦 𝐿,𝑎,𝑑,𝑚,𝑥,𝑦 𝑋,𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦, 𝑎) 𝐶(𝑎) 𝐷(𝑎, 𝑑) 𝑆(𝑎) 1 (𝑎, 𝑑) 𝐹(𝑎) 𝐺(𝑥, 𝑦, 𝑚, 𝑎, 𝑑) 𝑁(𝑎, 𝑑) 𝑊(𝑦, 𝑚, 𝑎, 𝑑) 𝑍(𝑎, 𝑑)

**Proof of Theorem dchrisum0lem1**
Step	Hyp	Ref	Expression
1		fzfid 14084	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		fzfid 14084	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∈ Fin)
3		fzfid 14084	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
4		elfznn 13655	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
5		elfzuz 13621	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
6	4, 5	anim12i 625	. . . . . 6 ⊢ ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
7	6	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
8		elfzuz 13621	. . . . . . 7 ⊢ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
9		elfznn 13655	. . . . . . 7 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
10	8, 9	anim12ci 626	. . . . . 6 ⊢ ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))))
11	10	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))))
12		eluzelz 12944	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → 𝑚 ∈ ℤ)
13	12	ad2antll 742	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℤ)
14	13	zred 12772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ)
15		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
16		2z 12697	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
17		rpexpcl 14191	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
18	15, 16, 17	sylancl 598	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
19	18	rpred 13133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ)
20	19	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ)
21		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℕ)
22	21	nnrpd 13131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ⁺)
23	14, 20, 22	lemuldivd 13182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
24	21	nnred 12319	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
25	15	rprege0d 13140	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
26		flge0nn0 13928	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
27		nn0p1nn 12614	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
28	25, 26, 27	3syl 19	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
29	28	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℕ)
30		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))
31		eluznn 13014	. . . . . . . . . . . 12 ⊢ ((((⌊‘𝑥) + 1) ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
32	29, 30, 31	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℕ)
33	32	nnrpd 13131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑚 ∈ ℝ⁺)
34	24, 20, 33	lemuldiv2d 13183	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 · 𝑑) ≤ (𝑥↑2) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
35	23, 34	bitr3d 284	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
36		rpcn 13100	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
37	36	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
38	37	sqvald 14254	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) = (𝑥 · 𝑥))
39	38	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) = (𝑥 · 𝑥))
40		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ⁺)
41	40	rpred 13133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ∈ ℝ)
42		reflcl 13904	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
43		peano2re 11454	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘𝑥) ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
44	41, 42, 43	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ∈ ℝ)
45		fllep1 13909	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
46	41, 45	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ ((⌊‘𝑥) + 1))
47		eluzle 12947	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) → ((⌊‘𝑥) + 1) ≤ 𝑚)
48	47	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((⌊‘𝑥) + 1) ≤ 𝑚)
49	41, 44, 14, 46, 48	letrd 11438	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑥 ≤ 𝑚)
50	41, 14, 40	lemul1d 13176	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 ≤ 𝑚 ↔ (𝑥 · 𝑥) ≤ (𝑚 · 𝑥)))
51	49, 50	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥 · 𝑥) ≤ (𝑚 · 𝑥))
52	39, 51	eqbrtrd 5126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ≤ (𝑚 · 𝑥))
53	20, 41, 33	ledivmuld 13186	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (((𝑥↑2) / 𝑚) ≤ 𝑥 ↔ (𝑥↑2) ≤ (𝑚 · 𝑥)))
54	52, 53	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ≤ 𝑥)
55		nnre 12311	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
56	55	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 𝑑 ∈ ℝ)
57	20, 32	nndivred 12361	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑚) ∈ ℝ)
58		letr 11375	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ ℝ ∧ ((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
59	56, 57, 41, 58	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ ((𝑥↑2) / 𝑚) ∧ ((𝑥↑2) / 𝑚) ≤ 𝑥) → 𝑑 ≤ 𝑥))
60	54, 59	mpan2d 707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑑 ≤ 𝑥))
61	35, 60	sylbid 243	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑑 ≤ 𝑥))
62	61	pm4.71rd 572	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
63		nnge1 12335	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → 1 ≤ 𝑑)
64	63	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → 1 ≤ 𝑑)
65		1re 11279	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
66		0lt1 11807	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
67	65, 66	pm3.2i 476	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ ∧ 0 < 1)
68	22	rpregt0d 13139	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ ℝ ∧ 0 < 𝑑))
69	18	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℝ⁺)
70	69	rpregt0d 13139	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2)))
71		lediv2 12176	. . . . . . . . . . . . . 14 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑑 ∈ ℝ ∧ 0 < 𝑑) ∧ ((𝑥↑2) ∈ ℝ ∧ 0 < (𝑥↑2))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
72	67, 68, 70, 71	mp3an2i 1495	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (1 ≤ 𝑑 ↔ ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1)))
73	64, 72	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ ((𝑥↑2) / 1))
74	20	recnd 11308	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑥↑2) ∈ ℂ)
75	74	div1d 12054	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 1) = (𝑥↑2))
76	73, 75	breqtrd 5130	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ≤ (𝑥↑2))
77		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → 𝑑 ∈ ℕ)
78		nndivre 12348	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((𝑥↑2) / 𝑑) ∈ ℝ)
79	19, 77, 78	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑥↑2) / 𝑑) ∈ ℝ)
80		letr 11375	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℝ ∧ ((𝑥↑2) / 𝑑) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
81	14, 79, 20, 80	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ≤ ((𝑥↑2) / 𝑑) ∧ ((𝑥↑2) / 𝑑) ≤ (𝑥↑2)) → 𝑚 ≤ (𝑥↑2)))
82	76, 81	mpan2d 707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) → 𝑚 ≤ (𝑥↑2)))
83	35, 82	sylbird 263	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) → 𝑚 ≤ (𝑥↑2)))
84	83	pm4.71rd 572	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ≤ ((𝑥↑2) / 𝑚) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
85	35, 62, 84	3bitr3d 312	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑)) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
86		fznnfl 13970	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
87	86	baibd 549	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
88	41, 21, 87	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ 𝑑 ≤ 𝑥))
89	79	flcld 13906	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ)
90		elfz5 13617	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
91	30, 89, 90	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
92		flge 13913	. . . . . . . . . 10 ⊢ ((((𝑥↑2) / 𝑑) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
93	79, 13, 92	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ ((𝑥↑2) / 𝑑) ↔ 𝑚 ≤ (⌊‘((𝑥↑2) / 𝑑))))
94	91, 93	bitr4d 285	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ↔ 𝑚 ≤ ((𝑥↑2) / 𝑑)))
95	88, 94	anbi12d 644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑑 ≤ 𝑥 ∧ 𝑚 ≤ ((𝑥↑2) / 𝑑))))
96	20	flcld 13906	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (⌊‘(𝑥↑2)) ∈ ℤ)
97		elfz5 13617	. . . . . . . . . 10 ⊢ ((𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)) ∧ (⌊‘(𝑥↑2)) ∈ ℤ) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
98	30, 96, 97	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
99		flge 13913	. . . . . . . . . 10 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑚 ∈ ℤ) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
100	20, 13, 99	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ≤ (𝑥↑2) ↔ 𝑚 ≤ (⌊‘(𝑥↑2))))
101	98, 100	bitr4d 285	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ↔ 𝑚 ≤ (𝑥↑2)))
102		fznnfl 13970	. . . . . . . . . 10 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
103	102	baibd 549	. . . . . . . . 9 ⊢ ((((𝑥↑2) / 𝑚) ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
104	57, 21, 103	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) ↔ 𝑑 ≤ ((𝑥↑2) / 𝑚)))
105	101, 104	anbi12d 644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) ↔ (𝑚 ≤ (𝑥↑2) ∧ 𝑑 ≤ ((𝑥↑2) / 𝑚))))
106	85, 95, 105	3bitr4d 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1)))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
107	106	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘((⌊‘𝑥) + 1))) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))))))
108	7, 11, 107	pm5.21ndd 382	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) ↔ (𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))))))
109		ssun2 4124	. . . . . . . 8 ⊢ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
110	28	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ ℕ)
111		nnuz 12973	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
112	110, 111	eleqtrdi 2870	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
113		dchrisum0lem1a 27776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))))
114	113	simprd 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥)))
115		fzsplit2 13651	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
116	112, 114, 115	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
117	109, 116	sseqtrrid 3973	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
118	117	sselda 3930	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
119		rpvmasum2.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
120		rpvmasum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
121		rpvmasum2.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
122		rpvmasum.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍)
123		rpvmasum2.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
124	123	ssrab3 4029	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
125		dchrisum0.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
126	124, 125	sselid 3928	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
127	126	eldifad 3910	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
128	127	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑋 ∈ 𝐷)
129		elfzelz 13625	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℤ)
130	129	adantl 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℤ)
131	119, 120, 121, 122, 128, 130	dchrzrhcl 27535	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
132		elfznn 13655	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℕ)
133	132	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℕ)
134	133	nnrpd 13131	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℝ⁺)
135	134	rpsqrtcld 15546	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℝ⁺)
136	135	rpcnd 13135	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℂ)
137	135	rpne0d 13138	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ≠ 0)
138	131, 136, 137	divcld 12062	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
139	4	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
140	139	nnrpd 13131	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
141	140	rpsqrtcld 15546	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℝ⁺)
142	141	rpcnne0d 13142	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
143	142	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
144	143	simpld 500	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ∈ ℂ)
145	143	simprd 501	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑑) ≠ 0)
146	138, 144, 145	divcld 12062	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
147	118, 146	syldan 603	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
148	147	anasss 472	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
149	1, 2, 3, 108, 148	fsumcom2 15907	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
150	149	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
151	65	a1i 11	. . 3 ⊢ (𝜑 → 1 ∈ ℝ)
152		2cn 12387	. . . . . . . 8 ⊢ 2 ∈ ℂ
153	15	rpsqrtcld 15546	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
154	153	rpcnd 13135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℂ)
155		mulcl 11255	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (√‘𝑥) ∈ ℂ) → (2 · (√‘𝑥)) ∈ ℂ)
156	152, 154, 155	sylancr 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (√‘𝑥)) ∈ ℂ)
157	141	rprecred 13144	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℝ)
158	1, 157	fsumrecl 15867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℝ)
159	158	recnd 11308	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) ∈ ℂ)
160	159, 156	subcld 11640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) ∈ ℂ)
161		2re 12386	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
162		dchrisum0.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
163		elrege0 13554	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
164	162, 163	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
165	164	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
166		remulcl 11256	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (2 · 𝐶) ∈ ℝ)
167	161, 165, 166	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
168	167	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℝ)
169	168, 153	rerpdivcld 13164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
170	169	recnd 11308	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
171	156, 160, 170	adddird 11305	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
172	156, 159	pncan3d 11643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)))
173	172	oveq1d 7423	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) · ((2 · 𝐶) / (√‘𝑥))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
174		2cnd 12390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
175	174, 154, 170	mulassd 11303	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))))
176	168	recnd 11308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℂ)
177	153	rpne0d 13138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ≠ 0)
178	176, 154, 177	divcan2d 12064	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥))) = (2 · 𝐶))
179	178	oveq2d 7424	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · ((√‘𝑥) · ((2 · 𝐶) / (√‘𝑥)))) = (2 · (2 · 𝐶)))
180	175, 179	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) = (2 · (2 · 𝐶)))
181	180	oveq1d 7423	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((2 · (√‘𝑥)) · ((2 · 𝐶) / (√‘𝑥))) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
182	171, 173, 181	3eqtr3d 2803	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))))
183	182	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))))
184		remulcl 11256	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ (2 · 𝐶) ∈ ℝ) → (2 · (2 · 𝐶)) ∈ ℝ)
185	161, 167, 184	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℝ)
186	185	recnd 11308	. . . . . 6 ⊢ (𝜑 → (2 · (2 · 𝐶)) ∈ ℂ)
187	186	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (2 · 𝐶)) ∈ ℂ)
188	160, 170	mulcld 11300	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
189		rpssre 13097	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
190		o1const 15754	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · (2 · 𝐶)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
191	189, 186, 190	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (2 · 𝐶))) ∈ 𝑂(1))
192		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))))
193	192	divsqrsum 27272	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟
194		rlimdmo1 15752	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ dom ⇝_𝑟 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
195	193, 194	mp1i 14	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥)))) ∈ 𝑂(1))
196	176, 154, 177	divrecd 12065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) = ((2 · 𝐶) · (1 / (√‘𝑥))))
197	196	mpteq2dva 5197	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))))
198	153	rprecred 13144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / (√‘𝑥)) ∈ ℝ)
199	167	recnd 11308	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝐶) ∈ ℂ)
200		rlimconst 15678	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (2 · 𝐶) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
201	189, 199, 200	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · 𝐶)) ⇝_𝑟 (2 · 𝐶))
202		sqrtlim 27263	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0
203	202	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / (√‘𝑥))) ⇝_𝑟 0)
204	168, 198, 201, 203	rlimmul 15779	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) · (1 / (√‘𝑥)))) ⇝_𝑟 ((2 · 𝐶) · 0))
205	197, 204	eqbrtrd 5126	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0))
206		rlimo1 15751	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ⇝_𝑟 ((2 · 𝐶) · 0) → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
207	205, 206	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · 𝐶) / (√‘𝑥))) ∈ 𝑂(1))
208	160, 170, 195, 207	o1mul2 15759	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
209	187, 188, 191, 208	o1add2 15758	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((2 · (2 · 𝐶)) + ((Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) − (2 · (√‘𝑥))) · ((2 · 𝐶) / (√‘𝑥))))) ∈ 𝑂(1))
210	183, 209	eqeltrd 2860	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ 𝑂(1))
211	158, 169	remulcld 11310	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
212	3, 147	fsumcl 15866	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
213	1, 212	fsumcl 15866	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
214	213	abscld 15573	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
215	211	recnd 11308	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℂ)
216	215	abscld 15573	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))) ∈ ℝ)
217	212	abscld 15573	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
218	1, 217	fsumrecl 15867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ ℝ)
219	1, 212	fsumabs 15935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
220	169	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
221	157, 220	remulcld 11310	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ∈ ℝ)
222	118, 138	syldan 603	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
223	3, 222	fsumcl 15866	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
224	223	abscld 15573	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
225		rpvmasum.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
226		rpvmasum2.1	. . . . . . . . . . 11 ⊢ 1 = (0_g‘𝐺)
227		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
228		dchrisum0.s	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
229		dchrisum0.1	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
230	120, 122, 225, 119, 121, 226, 123, 125, 227, 162, 228, 229	dchrisum0lem1b 27805	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))
231	224, 220, 141, 230	lediv1dd 13191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) ≤ (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)))
232	141	rpcnd 13135	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ∈ ℂ)
233	141	rpne0d 13138	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑑) ≠ 0)
234	223, 232, 233	absdivd 15592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))))
235	3, 232, 222, 233	fsumdivc 15919	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
236	235	fveq2d 6877	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
237	141	rprege0d 13140	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)))
238		absid 15430	. . . . . . . . . . . 12 ⊢ (((√‘𝑑) ∈ ℝ ∧ 0 ≤ (√‘𝑑)) → (abs‘(√‘𝑑)) = (√‘𝑑))
239	237, 238	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(√‘𝑑)) = (√‘𝑑))
240	239	oveq2d 7424	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (abs‘(√‘𝑑))) = ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)))
241	234, 236, 240	3eqtr3rd 2804	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) / (√‘𝑑)) = (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
242	170	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℂ)
243	242, 232, 233	divrec2d 12066	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((2 · 𝐶) / (√‘𝑥)) / (√‘𝑑)) = ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
244	231, 241, 243	3brtr3d 5135	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ ((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
245	1, 217, 221, 244	fsumle 15933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
246	157	recnd 11308	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑑)) ∈ ℂ)
247	1, 170, 246	fsummulc1 15918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
248	245, 247	breqtrrd 5132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
249	214, 218, 211, 219, 248	letrd 11438	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))))
250	211	leabsd 15549	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
251	214, 211, 216, 249, 250	letrd 11438	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
252	251	adantrr 730	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ≤ (abs‘(Σ𝑑 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑑)) · ((2 · 𝐶) / (√‘𝑥)))))
253	151, 210, 211, 213, 252	o1le 15787	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
254	150, 253	eqeltrrd 2861	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 {crab 3412 ∖ cdif 3895 ∪ cun 3896 ⊆ wss 3898 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5647 ‘cfv 6527 (class class class)co 7408 ℂcc 11169 ℝcr 11170 0cc0 11171 1c1 11172 + caddc 11174 · cmul 11176 +∞cpnf 11311 < clt 11314 ≤ cle 11315 − cmin 11512 / cdiv 11942 ℕcn 12304 2c2 12366 ℕ₀cn0 12575 ℤcz 12662 ℤ_≥cuz 12934 ℝ⁺crp 13089 [,)cico 13447 ...cfz 13608 ⌊cfl 13898 seqcseq 14112 ↑cexp 14172 √csqrt 15367 abscabs 15368 ⇝ cli 15618 ⇝_𝑟 crli 15619 𝑂(1)co1 15620 Σcsu 15820 Basecbs 17348 0_gc0g 17571 ℤRHomczrh 21766 ℤ/nℤczn 21769 DChrcdchr 27522

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-inf2 9620 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249 ax-addf 11250 ax-mulf 11251

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-ec 8697 df-qs 8701 df-map 8827 df-pm 8828 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9991 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-z 12663 df-dec 12784 df-uz 12935 df-q 13045 df-rp 13090 df-xneg 13210 df-xadd 13211 df-xmul 13212 df-ioo 13449 df-ioc 13450 df-ico 13451 df-icc 13452 df-fz 13609 df-fzo 13757 df-fl 13900 df-mod 13978 df-seq 14113 df-exp 14173 df-fac 14385 df-bc 14414 df-hash 14442 df-shft 15187 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-limsup 15605 df-clim 15622 df-rlim 15623 df-o1 15624 df-lo1 15625 df-sum 15821 df-ef 16200 df-sin 16202 df-cos 16203 df-pi 16205 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-starv 17404 df-sca 17405 df-vsca 17406 df-ip 17407 df-tset 17408 df-ple 17409 df-ds 17411 df-unif 17412 df-hom 17413 df-cco 17414 df-rest 17554 df-topn 17555 df-0g 17573 df-gsum 17574 df-topgen 17575 df-pt 17576 df-prds 17579 df-xrs 17635 df-qtop 17640 df-imas 17641 df-qus 17642 df-xps 17643 df-mre 17717 df-mrc 17718 df-acs 17720 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-mhm 18939 df-submnd 18940 df-grp 19108 df-minusg 19109 df-sbg 19110 df-mulg 19239 df-subg 19294 df-nsg 19295 df-eqg 19296 df-ghm 19389 df-cntz 19492 df-cmn 19957 df-abl 19958 df-mgp 20322 df-rng 20336 df-ur 20369 df-ring 20422 df-cring 20423 df-oppr 20528 df-dvdsr 20548 df-unit 20549 df-rhm 20663 df-subrng 20759 df-subrg 20783 df-lmod 21098 df-lss 21168 df-lsp 21208 df-sra 21409 df-rgmod 21410 df-lidl 21447 df-rsp 21448 df-2idl 21504 df-psmet 21631 df-xmet 21632 df-met 21633 df-bl 21634 df-mopn 21635 df-fbas 21636 df-fg 21637 df-cnfld 21640 df-zring 21714 df-zrh 21770 df-zn 21773 df-top 23173 df-topon 23190 df-topsp 23212 df-bases 23225 df-cld 23298 df-ntr 23299 df-cls 23300 df-nei 23377 df-lp 23415 df-perf 23416 df-cn 23506 df-cnp 23507 df-haus 23594 df-cmp 23666 df-tx 23842 df-hmeo 24035 df-fil 24126 df-fm 24218 df-flim 24219 df-flf 24220 df-xms 24600 df-ms 24601 df-tms 24602 df-cncf 25160 df-limc 26147 df-dv 26148 df-log 26847 df-cxp 26848 df-dchr 27523

This theorem is used by: dchrisum0lem3 27809

W3C validator